On quartic half-arc-transitive metacirculants

Abstract

Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m, n)-semiregular for some integers m ≥ 1, n ≥ 2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ m has at least one fixed vertex. A half-arc-transitive graph is a vertex-and edge-but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic halfarc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.Keywords: Graph; Metacirculant graph; Half-arc-transitive; Tightly attached; Automorphism group 1 Introductory and historic remarks Throughout this paper graphs are assumed to be finite and, unless stated otherwise, simple, connected and undirected (but with an implicit orientation of the edges when appropriate). For group-theoretic concepts not defined here we refer the reader to [4,9,32], and for graph-theoretic terms not defined here we refer the reader to [5].Given a graph X we let V (X), E(X), A(X) and AutX be the vertex set, the edge set, the arc set and the automorphism group of X, respectively. A graph X is said to be vertex-transitive, edge-transitive and arc-transitive if its automorphism group AutX acts transitively on V (X), E(X) and A(X), respectively. We say that X is half-arctransitive provided it is vertex-and edge-but not arc-transitive. More generally, by a half arc-transitive action of a subgroup G ≤ AutX on X we mean a vertex-and edge-but not arc-transitive action of G on X. In this case we say that the graph X is (G, 1 2 )-arc-transitive, and we say that the graph X is (G, 1 2 , H)-arc-transitive when it needs to be stressed that the vertex stabilizers G v (for v ∈ V (X)) are isomorphic to a particular subgroup H ≤ G. By a classical result of Tutte [30, 7.35, p.59], a graph admitting a half-arc-transitive group action is necessarily of even valency. A 1 Supported in part by "ARRS -Agencija za znanost Republike Slovenije", program no. P1-0285.

A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

“…A weak metacirculant is a graph whose automorphism group contains a vertex-transitive metacyclic group G, generated by ρ and σ, such that the cyclic group ρ is semiregular on the vertex-set of the graph, and is normal in G. This notion was introduced by Marušič andŠparl [22] and generalises that of a metacirculant introduced by Alspach and Parsons [2]. Metacirculants admitting 1 2 -arc-transitive groups of automorphisms were first investigated in [33].…”

“…Metacirculants admitting 1 2 -arc-transitive groups of automorphisms were first investigated in [33]. Recently, the interesting problem of classifying all 4-HATs that are weak metacirculants was considered in [22,35,36]. Such 4-HATs fall into four (not necessarily disjoint) classes (called Class I, Class II, Class III, and Class IV), depending on the structure of the quotient by the orbits of the semiregular element ρ.…”

“…After a short break, interest in GHATs picked up again in the 90s, largely due to a series of influential papers of Marušič concerning the GHATs of valence 4 (see [1,17,21,24], to list a few). For a nice survey of the topic, we refer the reader to [16], and for an overview of some more recent results, see [13,22].…”

Recent trends and future directions in vertex-transitive graphs

“…In particular, a classification of certain restricted families and various constructions of new families of such graphs together with some structural properties are known, see [11,29,41,71,74,75,84,85,86,87,88,98,99,112,115,116,117,118,123,129]. There are several approaches used in this respect, such as for example, investigation of the (im)primitivity of half-arc-transitive group actions on graphs, geometry related questions, and questions concerning classification for various restricted families of half-arc-transitive graphs.…”

Finite vertex-primitive edge-transitive metacirculants

“…Some special classes of metacirculants have been well-characterised, see [1,12,14] for edge-transitive circulants (that is, Cayley graphs of cyclic groups); [9,20,21] for 2-arc transitive dihedrants (that is, Cayley graphs of dihedral groups); [18,34] for half-arc-transitive metacirculants of prime-power order; [24,35] for half-arc-transitive metacirculants of valency 4. This paper is one of a series of papers to attack Problem A. A graph Γ is called vertex-primitive if AutΓ is a primitive permutation group on its vertex set.…”